Anomalous bulk-edge correspondence in continuous media

Abstract

Topology plays an increasing role in physics beyond the realm of topological insulators in condensed mater. From geophysical fluids to active matter, acoustics or photonics, a growing family of systems presents topologically protected chiral edge modes. The number of such modes should coincide with the bulk topological invariant (e.g. Chern number) defined for a sample without boundary, in agreement with the bulk-edge correspondence. However this is not always the case when dealing with continuous media where there is no small scale cut-off. The number of edge modes actually depends on the boundary condition, even when the bulk is properly regularized, showing an apparent paradox where the bulk-edge correspondence is violated. In this paper we solve this paradox by showing that the anomaly is due to {ghost} edge modes hidden in the asymptotic part of the spectrum. We provide a general formalism based on scattering theory to detect all edge modes properly, so that the bulk-edge correspondence is restored. We illustrate this approach through the odd-viscous shallow-water model and the massive Dirac Hamiltonian, and discuss the physical consequences.